I had time off when my kids were sick at times, my staff had time off when kids were sick but what is taking the mick is just expecting employers to have strategies for dealing with it without accepting that parents need to have strategies as well.
Three lots of absence in what six months for a 13 year old? That is alot, most teenagers can be left for at least part of the day. Maybe this time he can't but on all occasions? Most parents taking time off are for younger children. Perhaps the OPs employer thinks that have made alot of concessions in 13 years and they are expecting OP to be a bit more proactive. As I said before if you meet employers halfway you get a better response. If its always, "I'm taking the day off" then people will get tired of it. Certainly colleagues without children will.
As I said before if you come with an offer rather than a demand it goes along way e.g. I can't get in this morning but I'm hoping he might be feeling a bit better by lunchtime so if I can I will be in then. Can't you see why that would make a difference?
OP says its a one off but people are quoting previous incidents, she says she hasn't had time off in the past but lets face it that isn't going to be likely in 19 years is it? Particularly when that obviously includes a pregnancy. I haven't gone looking for previous posts so I have no idea if you have posted before but people do seem quite clear about it.
OP if you child has a health problem make sure your employer knows about it, if you can get in for part of a day then do and make sure your childless colleagues don't think you are taking advantage. It can cause alot of resentment and makes things harder for other parents.
Believe me that is constructive advice.